..so you are saying that a believer can't sin?
...
Perhaps you think that the word "sin" has some magical property or meaning?
Believer is too vague word, it can mean almost anything and I think it is possible that “believer” sins. For example, Bible tells that even demons believe and so they could be called “believers”. I think it does not mean they could not sin.
You believe that God is one. You do well. The demons also believe, and shudder.
James 2:19
Bible tells righteous person, children of God, do not sin.
He who does righteousness is righteous, even as he is righteous. He who sins is of the devil, for the devil has been sinning from the beginning. To this end the Son of God was revealed, that he might destroy the works of the devil. Whoever is born of God doesn't commit sin, because his seed remains in him; and he can't sin, because he is born of God. In this the children of God are revealed, and the children of the devil. Whoever doesn't do righteousness is not of God, neither is he who doesn't love his brother.
1 John 3:7-10
And eternal life is promised for righteous. So, if no one is really righteous, no one will get the life.
These will go away into eternal punishment, but the righteous into eternal life.
Mat. 25:46
For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord.
Romans 6:23
I believe there are people who will get the life, which means there are people who are righteous and don’t sin.
And I have understood righteous and sin are opposites. If person does something out of righteousness, it is righteous and not sin and if person does something in sin, it is sin. This means, the act itself is not necessary a sin, if it is not done in sinful state, if person is not unrighteous. Similarly, as the fruit of a tree is determined by the tree, also the actions are determined by the person. For example, an apple tree produces apples. It may be that some of them are rotten, but they are still apples. I think the same is with people. If person is righteous, he produces righteous fruit. And if person is unrighteous, in sin, he produces sin.
By their fruits you will know them. Do you gather grapes from thorns, or figs from thistles? Even so, every good tree produces good fruit; but the corrupt tree produces evil fruit. A good tree can’t produce evil fruit, neither can a corrupt tree produce good fruit. Every tree that doesn’t grow good fruit is cut down, and thrown into the fire. Therefore, by their fruits you will know them.
Matt. 7:16-20
Essentially, I think sin means that person has rejected God. After that he produces Godless fruit, which is also then a sin.